# Contrast-Independent Partially Explicit Time Discretizations for Quasi Gas Dynamics

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## Abstract

**:**

## 1. Introduction

- We use multiscale spatial decomposition that identifies fast and slow components of the solution.
- Proposed approach uses implicit treatment for a few degrees of freedom and the rest is treated explicitly. The resulting time constraint is independent of the contrast and scales as the coarse-mesh size.

## 2. Preliminaries

## 3. Partially Explicit Temporal Splitting Scheme

#### 3.1. Stability

**Lemma**

**1.**

**Proof.**

#### 3.2. Case ${V}_{H,1}$ and ${V}_{H,2}$ Are Orthogonal

**Lemma**

**2.**

**Proof.**

#### 3.3. Case ${V}_{H,1}$ and ${V}_{H,2}$ Are Non-Orthogonal

**Lemma**

**3.**

**Proof.**

#### 3.4. Remarks

#### 3.5. ${V}_{H,1}$ and ${V}_{H,2}$ Constructions

#### 3.5.1. CEM Method

#### 3.5.2. Construction of ${V}_{H,2}$

## 4. Numerical Result

- Implicit CEM. In this, we use only implicit method with fewer degrees of freedom (without additional degrees of freedom) and compute the error associated with the multiscale approach.
- Implicit CEM with additional basis functions. In this approach, we take into account additional degrees of freedom and handle them in implicit manner. The method is more expensive and corresponds to full implicit approach.
- Partial Explicit Splitting CEM. In our approach, we take into account additional degrees of freedom and handle them in explicit manner. One approach for additional degrees is presented in Section 3.5.2. Partial Explicit Splitting CEM and Implicit CEM have the same degrees of freedom, while partial Explicit Splitting CEM handles additional degrees of freedom explicitly.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**

**Left**: medium $\kappa $ with marker indicating three locations for comparison,

**Right**: source function f.

**Figure 8.**

**Top-Left**: Reference solution at different location.

**Top-Right**: Error of the Partially Explicit scheme at location $(0.375,0.375).$

**Bottom-Left**: Error of the Partially Explicit scheme at location $(0.5,0.375).$

**Bottom-Right**: Error of the Partially Explicit scheme at location $(0.625,0.375)$.

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**MDPI and ACS Style**

Chetverushkin, B.; Efendiev, Y.; Leung, W.T.
Contrast-Independent Partially Explicit Time Discretizations for Quasi Gas Dynamics. *Mathematics* **2022**, *10*, 576.
https://doi.org/10.3390/math10040576

**AMA Style**

Chetverushkin B, Efendiev Y, Leung WT.
Contrast-Independent Partially Explicit Time Discretizations for Quasi Gas Dynamics. *Mathematics*. 2022; 10(4):576.
https://doi.org/10.3390/math10040576

**Chicago/Turabian Style**

Chetverushkin, Boris, Yalchin Efendiev, and Wing Tat Leung.
2022. "Contrast-Independent Partially Explicit Time Discretizations for Quasi Gas Dynamics" *Mathematics* 10, no. 4: 576.
https://doi.org/10.3390/math10040576